Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model

Abstract

Spectral correlations are a powerful tool to study the dynamics of quantum many-body systems. For Hermitian Hamiltonians, quantum chaotic motion is related to random matrix theory spectral correlations. Based on recent progress in the application of spectral analysis to non-Hermitian quantum systems, we show that local level statistics, which probes the dynamics around the Heisenberg time, of a non-Hermitian q-body Sachdev-Ye-Kitev (nHSYK) model with N Majorana fermions, and its chiral and complex-fermion extensions, are also well described by random matrix theory for q > 2, while for q = 2 it is given by the equivalent of Poisson statistics. For that comparison, we combine exact diagonalization numerical techniques with analytical results obtained for some of the random matrix spectral observables. Moreover, depending on q and N , we identify 19 out of the 38 non-Hermitian universality classes in the nHSYK model, including those corresponding to the tenfold way.In particular, we realize explicitly 14 out of the 15 universality classes corresponding to nonpseudo-Hermitian Hamiltonians that involve universal bulk correlations of classes AI † and AII † , beyond the Ginibre ensembles. These results provide strong evidence of striking universal features in non-unitary many-body quantum chaos, which in all cases can be captured by nHSYK models with q > 2.

Lindbladian dissipation of strongly-correlated quantum matter

“…In generic (i.e., chaotic) closed manybody quantum systems, interactions may entail such a complex structure that the Hamiltonian behaves in several aspects like a large random matrix, as conjectured by Bohigas, Giannoni, and Schmit [4]. Extending this result to the dissipative realm is a fundamental problem that has attracted considerable attention recently [5][6][7][8][9][10][11]. Along similar lines, the past couple of years have seen the development of the (non-Hermitian) random matrix theory of Lindbladian dynamics [12][13][14][15][16][17][18][19][20].…”

“…Moreover, several experimental implementations have been proposed [65][66][67][68][69][70] and its practical and technological relevance has been highlighted [71][72][73][74][75][76]. Finally, non-Hermitian SYK models have also started gaining traction, with studies focusing on thermodynamics and wormhole physics [77,78], symmetries and universality [11], entanglement dynamics [79,80], and the effect of decoherence on quantum chaos [81,82].…”

“…respectively. Equations ( 9)- (11) can also be derived microscopically by tracing out the environment in a unitary system-plus-environment theory [104]. We further define the (mean-field) Green's function,…”

“…[15,21], is a natural next step. Finally, we mention the possibility of studying the symmetries of fermionic open quantum matter [11,108,109] in the context of the SYK model, for which a rich classification exists in the closed case [41][42][43].…”

Spacing distribution in the 2D Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at non-integer $β$

“…Likewise, based on the symmetry classes of Dirac operators [27], a classification of non-Hermitian random matrices was undertaken [28], that was recently revisited [29] and leads to 38 classes. Half of these classes have been identified in a non-Hermitian Sachdev-Ye-Kitaev model [30]. Based on heuristic arguments and numerics, it was found that only three distinct classes of 2D bulk statistics exist [15]: The Ginibre ensemble [6] and two further classes labelled AI † and AII † (in analogy to [26]), that possess additional symmetries under transposition.…”